Time-frequency Transfer Function – A New Tool for Modelling and Analysing Site Effects

Standard Fourier transfer function (modulus of the complex Fourier transfer function) is widely accepted and used for both theoretical/numerical analyses and analyses of real recordings, especially for identifying resonance peaks and/or the amplified frequency bands. This is because it characterizes (linear, time-invariant) transfer properties of the medium between a source and receiver in the frequency domain. Being, however, just an amplitude Fourier spectrum of the time-domain impulse response, it cannot alone provide any information on temporal development of the response, that is, the temporal distribution of the different reverberations that cause amplifications, and therefore on their origin.

However, it is reasonable to assume that the missing information could be useful for interpreting complicated resulting motions in local surface structures because they are often due to multiple wave reflections and transmissions, conversions, interference, diffraction, scattering and resonance.

Therefore, we introduce a new tool for analysing seismic response including its temporal development – a time-frequency transfer function, TFTF, based on the continuous wavelet transform of the impulse response. Modulus of TFTF provides information on the temporal development of frequency-dependent amplification and its duration, linking amplified frequency bands to specific arrivals and reverberation trains rather than to spectral peaks alone.

We present numerical examples for several sedimentary structures close to pure 1D layers, proving that TFTF is much more informative than the standard Fourier transfer function: in particular, it allows to identify late arrivals and long-lasting reverberations, providing a deeper insight on their physical origin.